Like so many things, there's how we think things should be, there's what the law seems to say, and then there's what happens to specific individuals at specific places with a specific instantiation of judges and bureaucrats. Local conditions always apply.
In the United States, and likely other areas, people mostly think you have to have at least a first and a last name. That belief is increasingly expressed through software and bureaucracy. If it ever became a legal issue, either by the mononymed person asserting their rights (or their parents doing it for them), or by some bureaucrat insisting that they toe the line, then it would end up depending on the whims of any involved judge and how badly you/they wanted whatever service you were trying to get (including, these days, buying an airplane ticket, where your SSN name has to match your Drivers License name and the name on the card that you buy the ticket with, because DHS).
As a parent, I would give my kid two names, first and last, because I wouldn't want to force this issue on them for mere iconoclasm. If you have some strong personal, religious or cultural beliefs, well, go ahead. But it's probably not your kid's fight.
I changed my name in the late eighties from First Middle Last, to First, through a name change order obtained through an application and then going before a judge. I'm certain that the local bureaucracy let it pass because of the nature of the local bureaucracy, and that other jurisdictions would not have allowed it, probably based purely on whim.
In the following twenty plus years, I have taken the position of being flexible. I still have an SSN card with just First on it, but I am no longer First according to SSN, because 9/11 and DHS. SSN now mails my documents to First Unk (or maybe the other way around, I don't care and don't remember).
I always try to be First, but often end up falling back to First First, or F. First. The IRS gladly accepts my check associated with my SSN number every year, and I always fill out their form with First in their last name field, which they have so far accepted.
I don't really care, I know who I am, and no one who knows me calls me anything but First. Sometimes new people call me Mr. First.
It's not something I care to fight about. I would never saddle my kid with that fight. Your choices are certainly your own.
When my son was born, about ten years after I changed my name, I named him HisFirstName First. So we would both have the same family name. :)